DESIGNING FOR LATERAL-TORSIONAL STABILITY IN WOOD MEMBERS

Transcription

1 DESIGNING FOR ATERA-TORSIONA STABIITY IN WOOD EBERS American Wood oncil TEHNIA REPORT 4 American Forest & Paper Association

2 The American Wood oncil (AW) is the wood prodcts division of the American Forest & Paper Association (AF&PA). AF&PA is the national trade association of the forest, paper and wood prodcts indstr, representing member companies engaged in growing, harvesting and processing wood and wood fiber, manfactring plp, paper and paperboard prodcts form both virgin and reccled fiber, and prodcing engineered and traditional wood prodcts. For more information see While ever effort has been made to insre the accrac of the information presented, and special effort has been made to assre that the information reflects the state-of-theart, neither the American Forest & Paper Association nor its members assme an responsibilit for an particlar design prepared from this pblication. Those sing this docment assme all liabilit from its se. opright 003 American Forest & Paper Association, Inc. American Wood oncil 9 th St., NW, Site 800 Washington, D awcinfo@afandpa.org

3 TEHNIA REPORT NO. 4 i TABE OF ONTENTS Page Table of ontents... i ist of Figres... ii ist of Tables... ii Nomenclatre... iv Part I: Basic oncept of ateral-torsional Bckling. Introdction.... Backgrond....3 Histor of the NDS Beam Stabilit Eqations Eqivalent oment ethod for Design of Wood Bending embers...8 Part II: Design ethod for ateral-torsional Bckling. onditions of ateral Spport...0. omparison of New alclation ethod and Experimental Data...3 Part III: Example Problems 3. Example 3. Beams Braced at Intermediate Points Example 3. ontinos Span Beams Example 3.3 antilevered Joists...7 References...9 Appendix A: Derivation of ateral-torsional Bckling Eqations... Appendix B: Derivation of 00 NDS Beam Stabilit Provisions...3 Appendix : Derivation of Rectanglar Beam Eqations...35 Appendix D: Graphical Representation of Wood Beam Stabilit Provisions...37 AERIAN FOREST & PAPER ASSOIATION

4 AERIAN WOOD OUNI ii DESIGNING FOR ATERA-TORSIONA STABIITY IN WOOD EBERS IST OF FIGURES Page Figre ateral-torsional Bckling of a Beam... Figre omparison of New alclation ethod with Test Data...3 Figre A ateral-torsional Bckling of a Beam (three views)... IST OF TABES Page Table omparison of Eqivalent oment Factor Approach with NDS Effective ength Provisions (for beams loaded throgh the netral axis)...8 Table b and k factors for Different oading onditions...

5 TEHNIA REPORT NO. 4 iii NOENATURE b = eqivalent moment factor for different loading and spport conditions e = load eccentricit factor fix = end fixit adjstment =.5 = beam stabilit factor w = warping constant d = depth of the member E = modls of elasticit E' 05 = adjsted modls of elasticit for bending abot the weak axis at the 5 th percentile G = torsional shear modls G' t05 = adjsted torsional shear modls at the 5 th percentile J = torsional constant for the section k = constant based on loading and spport conditions I x = moment of inertia in the strong-axis (x-direction) I = moment of inertia in the weak-axis (-direction) e = effective length which eqals the nbraced length adjsted for specific loading and spport conditions = nbraced length ' = adjsted moment resistance * = moment resistance for strong axis bending mltiplied b all applicable factors except f, V, and = minimm end moment = maximm end moment A = absolte vale of moment at /4 B = absolte vale of moment at / = absolte vale of moment at 3 /4 cr = critical bckling moment for specific loading and spport conditions max = absolte vale of maximm moment ocr = critical moment for the reference case (simpl-spported beam nder a constant moment) = cross-section slenderness adjstment = - I /I x AERIAN FOREST & PAPER ASSOIATION

6 TEHNIA REPORT NO. 4 Part I: Basic oncepts of ateral-torsional Bckling. INTRODUTION ateral-torsional bckling is a limit state where beam deformation incldes in-plane deformation, ot-of-plane deformation and twisting. The load casing lateral instabilit is called the elastic lateral-torsional bckling load and is inflenced b man factors sch as loading and spport conditions, member cross-section, and nbraced length. In the 00 and earlier versions of the National Design Specification (NDS ) for Wood onstrction [] the limit state of lateral torsional bckling is addressed sing an effective length format whereb nbraced lengths are adjsted to accont for load and spport conditions that inflence the lateral-torsional bckling load. Another common format ses an eqivalent moment factor to accont for these conditions. This report describes the basis of the crrent effective length approach sed in the NDS and smmarizes the eqivalent niform moment factor approach; provides a comparison between the two approaches; and proposes modification to NDS design provisions.. BAKGROUND Design of an nbraced beam reqires determination of the minimm moment which indces bckling. When a beam--one that is braced at its ends to prevent twisting at bearing points, bt which is otherwise nbraced along its length--is loaded in bending abot the strong axis so that lateral torsional bckling is indced, three eqilibrim eqations ma be written for the beam in its displaced position: one eqation associated with strong axis bending; one eqation associated with weak axis bending, and one eqation associated with twisting (see Figre ). Using these eqations of eqilibrim, a set of differential eqations can be developed which, when solved, provides the critical moment, cr. A closed-form soltion for the case of a simplspported beam nder a constant moment has been widel discssed and derivations can be fond in the literatre and are provided in Appendix A. The critical moment, Figre. ateral-torsional Bckling of a Beam ocr, for this reference case of a simpl-spported beam nder constant moment is given b: ocr EI GJ () AERIAN FOREST & PAPER ASSOIATION

7 DESIGNING FOR ATERA-TORSIONA STABIITY IN WOOD EBERS where: ocr E I G J = critical moment for the reference case (simpl-spported beam nder a constant moment) = nbraced length = modls of elasticit = moment of inertia in the weak-axis (-direction) = torsional shear modls = torsional constant for the section Determination of the torsional constant, J, is beond the scope of this docment. However, eqations for determining the vale of J for common shapes are provided in the literatre [][3][6]... Effect of Beam Slenderness Eqation was derived for slender beams where the in-plane flexral rigidit (EI x ) is mch larger than the ot-of-plane flexral rigidit (EI ). Federhofer and Dinnik [] fond this assmption to be conservative for less slender beams. To adjst for this conservatism in less slender beams where the flexral rigidities in each direction are of the same order of magnitde, Kirb and Nethercot [3] sggested dividing I b the term, where =-I /I x and I x is the moment of inertia abot the strong-axis... Effect of oading and Spport onditions Eqation applies to a single loading condition (simpl-spported beam nder a constant moment). an other loading and beam spport conditions are commonl fond in practice for which the magnitde of the critical moment needs to be determined. A closed-form soltion is not available for these conditions and differential eqations fond in the derivation mst be solved b other means inclding soltion b nmerical methods, infinite series approximation techniqes, finite element analsis, or empiricall. Soltions for common loading and beam spport conditions have been developed and are provided in design specifications and textbooks. Soltion techniqes sed to solve several common loading conditions and approximation techniqes sed to estimate adjstments for other loading conditions are discssed in more detail in Appendix A..... Adjsting for oading and Spport onditions Two methods are sed to adjst Eqation for different loading and spport conditions. The effective length approach and the eqivalent moment factor approach. Both of these methods adjst from a reference loading and spport condition.... Effective ength Approach The effective length approach adjsts the reference case critical moment, ocr, to other loading and spport conditions b adjsting the nbraced length to an effective length as provided in Eqation : cr EI GJ () e where: cr e = critical moment for specific loading and spport conditions = effective length which eqals the nbraced length adjsted for specific loading and spport conditions AERIAN WOOD OUNI

8 TEHNIA REPORT NO Eqivalent oment Factor Approach The eqivalent moment factor approach adjsts the reference case critical moment, ocr, to other loading and spport conditions sing an eqivalent moment factor, b, as provided in Eqation 3: where: b = eqivalent moment factor (3) cr b ocr Prior to 96, the steel indstr sed an effective length approach to distingish between different loading and spport conditions. In 96, the American Institte of Steel onstrction (AIS) adopted the eqivalent moment factor approach. This approach expanded application of beam stabilit calclations to a wider set of loading and spport conditions. The empirical eqation sed for design of steel beams when estimating the eqivalent moment factor, b, has changed over the last 40 ears. The original eqation for estimating b was: where: b = minimm end moment = maximm end moment (4) Eqation 4 provides accrate estimates of b in cases where the slope of the moment diagram is constant (a straight line) between braced points bt does not appl to beams bent in reverse crvatre. For this reason, it is onl recommended for / ratios which are negative (with single crvatre i.e. / < 0) [4]. Eqation 5 was developed for cases where slope of the moment diagram is constant between braced points and better matches the theoretical soltion for beams bent in single and reversed crvatre [4]. This eqation is of the form: b.5 (5) To address limitations of Eqations 4 and 5 and to avoid misapplication of these eqations, Kirb and Nethercot [3] proposed the following empirical eqation which ses a moment-fitting method to estimate b : max b (6) where: A = absolte vale of moment at /4 B = absolte vale of moment at / A B max AERIAN FOREST & PAPER ASSOIATION

9 4 DESIGNING FOR ATERA-TORSIONA STABIITY IN WOOD EBERS = absolte vale of moment at 3 /4 max = absolte vale of maximm moment The following modified version of the Kirb and Nethercot eqation appeared in AIS s 993 oad and Resistance Factored Design Specification for Strctral Steel Bildings [4]. This eqation provides a lower bond across the fll range of moment distribtions in beam segments between points of bracing: b.5 max (7) max A B..3 Effect of oad Position Relative to Netral Axis Eqation assmes that loading is throgh the netral axis of the member. In cases where the load is above the netral axis at an nbraced location, an additional moment is indced de to displacement of the top of the beam relative to the netral axis. This additional moment de to torsional eccentricit redces the critical moment. Similarl, a load applied below the netral axis increases the critical moment. The design literatre provides several means of addressing the effects of load position [][5][6][0]...3. Adjsting for oad Position Relative to Netral Axis There are generall two methods sed to adjst Eqation for load position relative to the netral axis. One method adjsts the effective length previosl discssed in... The second method ses a load eccentricit factor, e. Derivation of this factor has been reviewed in the literatre [][0] and is provided in Appendix A.. The eccentricit factor takes the form of: (8) e where: e k d kd EI (9) GJ = load eccentricit factor = constant based on loading and spport conditions = depth of the member Flint [0] demonstrated nmericall, that Eqation 9 is accrate for a single concentrated load case where lies between and +.7 (negative vales indicate loads applied below the netral axis and positive vales indicate loads applied above the netral axis). imiting to a maximm vale of.7 limits e to a minimm vale of Non-Rectanglar Beams Eqation nder-predicts the critical moment for beams with non-rectanglar cross-sections, sch as I-beams. For more accrate predictions of the critical moment for simpl spported nonrectanglar beams nder niform moment, an additional term can be added to accont for increased torsional resistance provided b warping of the cross-section []: AERIAN WOOD OUNI

10 TEHNIA REPORT NO. 4 5 ocr E EI GJ I w (0) where: w = warping constant Procedres for calclating the warping constant, w, are beond the scope of this docment; however, procedres exist in the literatre [][6]. For man cross-sections, Eqation 0 is overl complicated and can be simplified b conservativel ignoring the increased resistance provided b warping...5 antilevered and ontinos Beams rrentl, there is a lack of specific lateral-torsional bckling design provisions for cantilevered and continos beams. For cantilevered beams, direct soltions for a few cases have been developed and are available in the literatre [][3][4][6][] and have been reprinted in Tables and. However, for other load cases of cantilevered beams which are nbraced at the ends, approximation eqations, sch as Eqation 7, do not appl. For these cases, designers have sed conservative vales for b and k. Derivation of b and k vales for additional cantilevered load cases is beond the scope of this report. Provisions in this report also appl to continos beams. However, since there are varios levels of bracing along the compression edge of continos beams and at interior spports, appling beam stabilit provisions ma be confsing. As a reslt, most continos beams are designed with additional fll-height bracing at points of interior bearing. To clarif how to se the provisions in this report to design continos beams, it mst be clear that the derivation of Eqation assmes that there is no external lateral load applied abot the weak axis of the beam throgh bracing, spports, or otherwise. Also, it assmes that spports at the ends of members are laterall braced. Additional considerations are reqired for continos beams. A continos beam can be assmed to be braced adeqatel at interior spports if provisions are made to prevent displacement of the tension edge of the beam relative to the compression edge. Sch provisions inclde providing fll-height blocking, providing both tension and compression edge sheathing at the interior spports, or having one edge braced b sheathing and the other edge braced b an adeqatel designed wall. Otherwise, the designer has to se jdgment as to whether to consider the beam to have an nbraced length eqal to the fll length of the member between points of bracing, an nbraced length measred from the inflection point, or an nbraced length that is between these two conditions. The designer s jdgment shold consider the sheathing stiffness and the method of attachment to the beam. A beam that is abot to ndergo lateral torsional bckling develops bending moments abot its weak axis and these moments can extend beond inflection points associated with strong axis bending. Frther research is being considered to qantif the roles that tension side bracing and inflection points have on the tendenc for lateral torsional bckling...6 Beam Bracing Derivation of eqations in this report are based on the assmption that member ends at bearing points are braced to prevent rotation (twisting), even in members otherwise considered to be nbraced. In addition, effective beam bracing mst prevent displacement of the beam AERIAN FOREST & PAPER ASSOIATION

11 6 DESIGNING FOR ATERA-TORSIONA STABIITY IN WOOD EBERS compression edge relative to the beam tension edge (i.e. twisting of the beam). There are two tpes of beam bracing, lateral and torsional. ateral bracing, sch as sheathing attached to the compression edge of a beam, is sed to minimize or prevent lateral displacement of the beam, thereb redcing lateral displacement of the compression edge of the beam relative to the tension edge. Torsional bracing, sch as cross-bridging or blocking, is sed to minimize or prevent rotation of the cross-section. Both bracing tpes can be designed to effectivel control twisting of the beam; however, steel bracing sstems that provide combined lateral and torsional bracing, sch as a concrete slab attached to a steel beam compression edge with shear stds, have been shown to be more effective than either lateral or torsional bracing acting alone (tton and Trahair [7a]) (Tong and hen [7b]). Design of lateral and/or torsional bracing for wood beams is crrentl beond the scope of this docment; however, frther research is being considered to qantif and develop design provisions for lateral and torsional bracing sstems. Gidance for steel bracing sstems is provided in Yra [5] and Galambos [6]...6. ateral Bracing There are for general tpes of lateral bracing sstems; continos, discrete, relative, and leanon. The effectiveness and design of lateral bracing is a fnction of bracing tpe, brace spacing, beam cross-section and indced moment. ateral bracing is most effective when attached to the compression edge and has little effect when bracing the netral axis (Yra [5])...6. Torsional Bracing There are two general tpes of torsional bracing; continos and discrete. The effectiveness and design of torsional bracing is significantl affected b the cross-sectional resistance to twisting..3 HISTORY OF THE NDS BEA STABIITY EQUATIONS In earl editions of the NDS, beam stabilit was addressed sing prescriptive bracing provisions. The effective length approach (see...) for design of bending members was introdced in the 977 NDS. Basic load and spport cases addressed in the 977 NDS were as follows:. A simpl-spported beam with: - niforml-spported load nbraced along the loaded edge - a concentrated load at midspan nbraced at the point of load - eqal end moments ( opposite rotation ). A cantilevered beam with: - niforml distribted load- nbraced along the loaded edge - concentrated load at the free-end nbraced at the point of load 3. A conservative provision for all other load and spport cases Provisions for these load and spport conditions were derived from a report b Hoole and adsen [7]. Hoole and adsen sed a std b Flint [0] to derive simple effective length eqations to adjst the nbraced lengths for the above mentioned load cases and loads applied at the top of beams. In addition, Hoole and adsen recommended that the nbraced length be increased 5% to accont for imperfect torsional restraint at spports. Derivation of these provisions and comparisons are presented in Appendix B. AERIAN WOOD OUNI

12 TEHNIA REPORT NO. 4 7 In 99, the NDS provisions were expanded to cover a wider range of load cases based on a std b Hoole and Devall [8]. In addition, some load cases were changed [9]..3. omparison Between e Approach and b Approach In the NDS provisions which se an effective length approach to adjst for different load and spport conditions, the bckling stress, F be, is calclated sing a modified form of Eqation which has been simplified for rectanglar beams (see Appendix B). The eqation was derived for a simpl-spported beam with a concentrated load at midspan - nbraced at point of load, as provided in Eqation. Additional loading conditions were derived and the effective length eqations were adjsted to this reference condition. cr.40ei () e where: e =.37, for a simpl-spported beam with a concentrated load at midspan - nbraced at point of load For the reference condition (simpl-spported beam with a constant moment) in Eqation, the NDS effective length eqation is: e =.84. The eqivalent moment factor approach and the NDS effective length approach can be compared b recognizing that, for the reference condition, b =.0 and e =.84. Therefore, b =.84 / e. Solving for e ields:. 84 e () b Table compares effective lengths derived sing Eqations 7 and and effective lengths tablated in the 00 NDS. It shold be noted that some of the NDS effective length eqations are inconsistent with estimates from Eqations 7 and. In cases where a load can be applied on the compression edge of a beam, NDS Table assmes that the beam is alwas loaded on the compression edge. For cases not explicitl covered in Table of the NDS, the most severe effective length assmption applies (i.e..84 < e <.06 ). In addition, Hoole and Dvall assmed that effective lengths for beams loaded at mltiple points cold be estimated b interpolating between the single concentrated load case and the niform load case. This assmption is inconsistent with assmptions sed in the derivation of the eqations and the eqivalent moment factors. AERIAN FOREST & PAPER ASSOIATION

13 8 DESIGNING FOR ATERA-TORSIONA STABIITY IN WOOD EBERS Table. omparison of Eqivalent oment Factor Approach with NDS Effective ength Provisions (for beams loaded throgh the netral axis) oading ondition ateral oad Eqivalent oment Factor Approach b (Eqn. 7) Single Span Beams Effective ength, e (Eqn. ) 00 NDS Table Effective ength, e (NDS tablated vales) Uniforml distribted load No.3 e =.63 e = d oncentrated center No.35 e =.37 e = d Yes.67 e =.0 e =. Two eqal conc. /3 points No.4 e =.6.84 < e <.06 Yes.00 e =.84 e =.68 Three eqal conc. /4 points No.4 e =.6.84 < e <.06 Yes. e =.66 e =.54 For eqal conc. /5 points No.4 e =.6.84 < e <.06 Yes.00 e =.84 e =.68 Five eqal conc. /6 points No.4 e =.6.84 < e <.06 Yes.05 e =.75 e =.73 Six eqal conc. /7 points No.3 e = < e <.06 Yes.00 e =.84 e =.78 Seven eqal conc. /8 points No.3 e = < e <.06 Yes.03 e =.79 e =.84 Eight or more eqal conc. eqal spacings No.3 e = < e <.06 Yes.00 e =.84 e =.84 Eqal end moments (opposite rotation) -.00 e =.84 e =.84 Eqal end moments (same rotation) -.30 e = < e <.06 antilever Beams Uniforml distribted load No.05 e = 0.90 e = d oncentrated nbraced end No.8 e =.44 e = d d = depth of the member.4 EQUIVAENT OENT ETHOD FOR DESIGN OF WOOD BENDING EBERS Design provisions for lateral-torsional bckling of wood bending members are contained in the general design provisions for bending in the NDS [] and ASE 6 [4]. In the NDS, an adjstment is made to the bending stress, F b *, b appling a beam stabilit factor,, that acconts for the ratio of the bckling stress, F be, to the bending stress, F b *. Adjstments for loading, spport, and load eccentricit conditions are made to F be throgh se of tablated effective lengths, e (see...). AERIAN WOOD OUNI

14 TEHNIA REPORT NO. 4 9 In ASE 6, lateral-torsional bckling is addressed in a similar manner. An adjstment is made to the strong-axis bending moment, x *, b appling a beam stabilit factor,, that acconts for the ratio of the elastic lateral bckling moment, e, to the strong-axis bending moment, x *. Note that the elastic lateral bckling moment, e, is eqivalent to the critical moment, cr, derived in earlier sections of this docment. For prismatic beams, adjstments for loading, spport, and load eccentricit conditions are made to e throgh the se of tablated effective lengths, e. For non-prismatic beams, adjstments for loading and spport conditions are made sing eqivalent moment factors, b (see...), Vales for b are calclated sing Eqation 4 (see... for limitations on se of Eqation 4). A new, more comprehensive method for design of wood bending members is provided in Part II of this docment. The new method ses the ASE 6 format of appling a beam stabilit factor,, to the strong-axis bending moment, x * to accont for the ratio cr (also known as the elastic lateral bckling moment, e ), to x *. This method incorporates the se of eqivalent moment factors (...) and load eccentricit factors (see..3.) to accont for the effects of loading, spport, and load eccentricit effects on the critical moment, cr. AERIAN FOREST & PAPER ASSOIATION

15 0 DESIGNING FOR ATERA-TORSIONA STABIITY IN WOOD EBERS Part II: Design ethod for ateral-torsional Bckling. ONDITIONS OF ATERA SUPPORT.. General When a bending member is bent abot its strong axis and the moment of inertia abot the strong axis (I x ) exceeds the moment of inertia abot the weak axis (I ), the weak axis shall be braced in accordance with.. or the moment capacit in the strong axis direction shall be redced per Bracing of ompression Edge The compression edge of a bending member shall be braced throghot its length to prevent lateral displacement and the ends at points of bearing shall have lateral spport to prevent rotation. The adjsted moment resistance, ', abot the strong-axis of a member laterall braced as noted above is: where: * (3) ' = adjsted moment resistance * = moment resistance for strong axis bending mltiplied b all applicable factors except f, V, and...3 oment Resistance of embers Withot Fll ateral Spport The adjsted moment resistance abot the strong axis of members that are not laterall braced for their fll length or a portion thereof is: * (4) where: = beam stabilit factor (..3.)..3. Beam Stabilit Factor The beam stabilit factor,, shall be calclated as: b.90 b.90 b 0.95 (5) where: cr b * cr = critical bckling moment (6) When the volme effect factor, V, is less than.0, the lesser of or V shall be sed to determine the adjsted moment resistance. AERIAN WOOD OUNI

16 TEHNIA REPORT NO ritical Bckling oment The critical bckling moment, cr, shall be calclated as: cr b fix e E 05 I G 05 J where: b = eqivalent moment factor for different loading and spport conditions (..3.3) e = load eccentricit factor for top-loaded beams (..3.4) fix = end fixit adjstment (see.3) =.5 = nbraced length E' 05 = adjsted modls of elasticit for bending abot the weak axis at the fifth percentile I = weak axis moment of inertia G' t05 = adjsted torsional shear modls at the fifth percentile J = torsional constant = cross-section slenderness adjstment = - I /I x For rectanglar bending members, the critical bckling moment shall be permitted to be calclated as (see Appendix. for derivation): t (7) cr.3 E b e 05 I (8)..3.3 Eqivalent oment Factor The eqivalent moment factor, b, for different loading and spport conditions shall be taken from Table or, for beam segments between points of bracing, calclated as: where:.5 max b 3 A 4 B 3.5 (9) max max = absolte vale of maximm moment A = absolte vale of moment at /4 B = absolte vale of moment at / = absolte vale of moment at 3/4 For all loading and spport conditions, b is permitted to be conservativel taken as oad Eccentricit Factor for Beams oaded on the ompression-side When a bending member is loaded from the tension-side or throgh the netral axis, e =.0 which is conservative for cases where the bending member is loaded from the tension-side [][7][0]. When a bending member is loaded from the compression-side of the netral axis and is braced at the point of load, e =.0. When a bending member is loaded from the compressionside of the netral axis and is nbraced at the point of load, the load eccentricit factor shall be calclated as: AERIAN FOREST & PAPER ASSOIATION

17 DESIGNING FOR ATERA-TORSIONA STABIITY IN WOOD EBERS where: (0) e kd E 05I () G J t05 k = constant based on loading and spport conditions (Table ). For all loading cases, k can conservativel be taken as.7. The minimm vale of e need not be taken as less than 0.7. For rectanglar bending members, shall be permitted to be calclated as (see Appendix. for derivation):.3kd () Table. b and k Factors for Different oading onditions oading ondition aterall point of loading aterall point of loading b b k Single Span Beams oncentrated center Two eqal conc. /3 points Three eqal conc. /4 points For eqal conc. /5 points Five eqal conc. /6 points Six eqal conc. /7 points Seven eqal conc. /8 points Eight or more eqal conc. eqal spacings Uniforml distribted load Eqal end moments (opposite rotation) Eqal end moments (same rotation) antilever Beams oncentrated end Uniforml distribted load The nbraced length,, in these cases is zero; therefore, the beam is considered fll-braced with =.0. AERIAN WOOD OUNI

18 TEHNIA REPORT NO Special Design onsiderations for antilevered Beams For cantilevered beams, direct soltions for two cases have been developed and are provided in Table ; however, for other load cases of cantilevered beams which are nbraced at the ends, Eqation 8 does not appl. For these cases, designers are reqired to se conservative vales of b =.0 and k = Special Design onsiderations for ontinos Beams For continos beams, vales for b and k var depending on the location of bracing. A continos beam can be assmed to be adeqatel braced at an interior spport if fll-height blocking or tension and compression edge bracing at spports are sed to prevent displacement of the tension edge of the beam relative to the compression edge. Otherwise the beam shold be designed as nbraced for the fll length of the beam between points of bracing.. OPARISON OF NEW AUATION ETHOD AND EXPERIENTA DATA In 96, Hoole and adsen tested 33 beams with varios b/d ratios, loading conditions, and spport conditions [7]. Using behavioral eqations taken from the design procedre otlined in., the Hoole and adsen data was analzed. A comparison of the predicted bending strength with the observed bending strength is plotted in Figre Predicted Bending Strength (psi) Observed Bending Strength (psi) Figre. omparison of New alclation ethod with Test Data AERIAN FOREST & PAPER ASSOIATION

19 4 DESIGNING FOR ATERA-TORSIONA STABIITY IN WOOD EBERS Part III: Example Problems Example problems are based on the application of the new beam stabilit provisions in Part II. Example 3. - Beams Braced at Intermediate Points Problem Statement: Design a gllam beam to be sed as a ridge beam braced with roof joists spaced at 4 feet on center along its length. Given: The span of the ridge beam,, is 8 feet. The design loads are w dead = 75 plf and w snow = 300 plf. The roof joists are attached to the sides of the ridge beam providing adeqate bracing of the beam; therefore, for calclating the beam bckling moment the nbraced length of the beam,, can be assmed to be 48 inches. alclate the maximm indced moment: max = (w dead +w snow ) / 8 = (375)(8 )/8 = 36,750 ft-lb Select a ½" x " 4F-V4 Doglas fir gllam beam: F bx =,400 psi E =,600,000 psi alclate properties: * F bx = F bx D t f i = (,400)(.5)(.0)(.0)(.0)(.0) =,760 psi E' 05 = E (-.645OV E )(Adjst to shear-free E)/(Safet Factor) = (,600,000)[-.645(0.)](.05)/(.66) =,600,000/.93 = 89,000 psi S x = bd /6 = (.5)( )/6 = 83.8 in. 3 I x = bd 3 / = (.5)( 3 )/ = 99 in. 4 I = db 3 / = ()(.5 3 )/ = 7.34 in. 4 V = (/) /x (/d) /x (5.5/b) /x where x = 0 (/8) 0. (/) 0. (5.5/.5) 0. = alclate bckling moment capacit: b =.00 (assming 6 roof joists, from Table ) e =.00 (assming braced at point of load) cr =.3 b e E 05 ' I / =.3(.0)(.0)(89,000)(7.34)/(48) = 63,800 in-lb or 5,50 ft-lb alclate allowable bending moment: * = F * bx S x = (,760)(83.8) = 507,300 in-lb or 4,50 ft-lb b = cr / * = 5,50/4,50 =. b.9 b.9 b ' = * (lesser of V or ) = (4,50)(0.88) = 37, 50 ft-lb Strctral heck: ' max 37,50 ft-lb 36,750 ft-lb AERIAN WOOD OUNI

20 TEHNIA REPORT NO.4 5 Example 3. - ontinos Span Beams Problem Statement: Determine b, k, and vales for a two-span continos beam sbjected to a niforml distribted gravit load. ompare these vales for the following bracing conditions:. Top edge nbraced, interior bearing point braced.. Top edge nbraced, interior bearing point nbraced. Given: The beam has two eqal spans of length,. The vale for b can be calclated for each case sing Eqation 8: b 3 A B max.5 max ase : The top edge of the beam is in compression from the end to the inflection point of the beam (0.75) and is nbraced. The interior bearing point is braced relative to the top tension edge of the beam. Therefore, this case has two nbraced lengths,, each spanning from the end of the beam to the interior spport ( = ). Since a vale for k has not been derived, assme k=.7. The qarter-point moments for this case are as follows: Top edge nbraced Interior bearing provides lateral bracing max = interior bearing A = 0.065w at x=0.5 B = 0.065w at x=0.5 = at x=0.75 b 3(0.065w ) 4(0.065w.5(0.50w ) ) 3(0.0000w ).5(0.50w ase. Top edge nbraced, interior bearing point braced.08 ) AERIAN FOREST & PAPER ASSOIATION

21 6 DESIGNING FOR ATERA-TORSIONA STABIITY IN WOOD EBERS ase : The top edge of the beam is in compression from the end to the inflection point of the beam (0.75) and is nbraced. The interior bearing point is nbraced. Therefore, this case has one nbraced length,, which spans between the ends of the beam ( = ). Since a vale for k has not been derived, assme k=.7. The qarter-point moments for this case are as follows: Top edge nbraced Interior bearing does not provide lateral bracing max = interior bearing A = 0.065w at x=0.5 B = -0.50w at x= = 0.065w at x=.5 ase. Top edge nbraced, interior bearing point nbraced b 3(0.065w ) 4(0.50w.5(0.50w ) ) 3(0.065w ).5(0.50w.3 ) Note: This example assmes that there are no external lateral forces being resisted b the beam and/or interior spport. If lateral forces are being resisted b the beam in the weak axis direction, a different analsis inclding the effects on beam stabilit, shold be considered. AERIAN WOOD OUNI

22 TEHNIA REPORT NO.4 7 Example antilevered eiling Joists Problem Statement: Determine the impact of bracing the end of a cantilevered ceiling joist that is nbraced on the compression (bottom) edge. Given: A ceiling is designed with a cantilevered span of 4 feet. The beam is laterall braced at the spporting walls. The cantilever is spporting a concentrated load at the end of the cantilever. Design the cantilever with # Sothern pine x joists spaced at feet on center. From Table 4B of the NDS Spplement: F bx = 975 psi E =,600,000 psi alclate properties: * F bx = F bx D t F f i r = (975)(.0)(.0)(.0)(.0)(.0)(.0)(.5) =, psi E' 05 = E (-.645OV E )(Adjst to shear-free E)/(Safet Factor) = (,600,000)[-.645(0.5)](.03)/(.66) =,600,000/.74 = 584,500 psi S x = bd /6 = (.5)(.5 )/6 = 3.64 in. 3 I x = bd 3 / = (.5)(.5 3 )/ = 78.0 in. 4 I = db 3 / = (.5)(.5 3 )/ = 3.64 in. 4 Withot End Bracing alclate bckling moment capacit: b =.8 k =.0 (from Table ) =.3kd/ =.3(.0)(.5)/48 = e = ( + ) = ( ) = cr =.3 b e E 05 ' I / =.3(.8)(0.7407)(584,494)(3.64)/48 = 47,487 in-lb or 3,957 ft-lb alclate allowable bending moment: x * = F bx * S x = (,)(3.64) = 35,468 in-lb or,956 ft-lb b = cr / x * = 3,957/,956 = x ' = x * = (,956)(0.905) =,676 ft-lb. AERIAN FOREST & PAPER ASSOIATION

23 8 DESIGNING FOR ATERA-TORSIONA STABIITY IN WOOD EBERS With End Bracing alclate bckling moment capacit: b =.67 (from Table ) e =.00 (assming braced at point of load) cr =.3 b e E 05 ' I / =.3(.67)(.00)(584,494)(3.64)/48 = 83,644 in-lb or 6,970 ft-lb alclate allowable bending moment: x * = F bx * S x = (,)(3.64) = 35,468 in-lb or,956 ft-lb b = cr / x * = 6,970/,956 = x ' = x * = (,956)(0.966) =,857 ft-lb. Note: This example assmes that there are no external lateral forces being resisted b the beam and/or interior spport. If lateral forces are being resisted b the beam in the weak axis direction, a different analsis inclding the effects on beam stabilit shold be considered. AERIAN WOOD OUNI

24 TEHNIA REPORT NO. 4 9 References. National Design Specification (NDS) for Wood onstrction. American Forest & Paper Association Timoshenko, S. Theor of Elastic Stabilit. First Edition. cgraw-hill Book o., Kirb, P. A. and Nethercot, D. A. Design for Strctral Stabilit. onstrado onographs, Granada Pblishing, Sffolk, UK, oad and Resistance Factor Design Specification for Strctral Steel Bildings, AIS anal of Steel onstrction, Second Edition. American Institte of Steel onstrction Galambos, T.V. Gide to Stabilit Design riteria for etal Strctres. John Wile & Sons, Inc hen, W.F. and i, E.. Strctral Stabilit Theor and Implementation. Prentice Hall Hoole, R. F. and adsen, B. ateral Bckling of Gled aminated Beams. Jornal of Strctral Engineering Division, ASE, Vol. 90, No. ST3. Jne Hoole, R. F. and Devall, R. H. ateral Bckling of Simpl Spported Gled aminated Beams. Universit of British olmbia NDS ommentar. American Forest & Paper Association Flint, A. R. The ateral Stabilit of Unrestrained Beams. Engineering, Vol 73, 95, Nethercot, D. A. and Rocke, K.. A nified approach to the elastic lateral bckling of beams. The Strctral Engineer. Vol. 49, No. 7, Jl 97, Salvadori,. G. ateral bckling of I-beams. ASE Transaction Vol. 0, 955, AERIAN FOREST & PAPER ASSOIATION

25 0 DESIGNING FOR ATERA-TORSIONA STABIITY IN WOOD EBERS AERIAN WOOD OUNI

26 AERIAN FOREST & PAPER ASSOIATION TEHNIA REPORT NO. 4 Appendix A Derivation of ateral-torsional Bckling Eqations A. GENERA An nbraced beam loaded in the strong-axis is free to displace and rotate as shown in Figre A. This displacement and rotation can be represented b three eqations of eqilibrim which eqate the externall applied moment to the internal resisting moment: d v. moment in the strong-axis direction: ext EI x dz (A). moment in the weak-axis direction: ext EI d dz (A) 3. twisting moment: d dz ext GJ d dz (A3) Figre A. ateral-torsional Bckling of a Beam Eqation A describes the strong axis bending behavior of a beam and, for small displacements, is independent of Eqations A and A3. Eqations A and A3 are copled. B differentiating Eqation A3 with respect to z and sbstitting the reslt into Eqation A, the two eqations can be combined to form the following second-order differential eqation: d ext 0 dz EI GJ (A4) A.. OSED-FOR SOUTION If a beam has a constant cross-section and a constant indced moment, o, along the nbraced length,, there is a closed-form soltion to Eqation A4. B defining a constant k = o /EI GJ, Eqation A4 can be rewritten as:

27 AERIAN WOOD OUNI DESIGNING FOR ATERA-TORSIONA STABIITY IN WOOD EBERS d k dz 0 (A5) Eqation A5 is a second-order linear differential eqation with a general soltion of: Asin kz Bcoskz (A6) Given that the rotation of the beam at the ends is prevented de to end restraint, these bondar conditions can be expressed as (0) = 0 and () = 0. Sbstitting the first condition into Eqation A6, it can be seen that B = 0. Sbstitting the second condition into Eqation A6 provides the following soltion: Asin k 0 (A7) To provide a nontrivial soltion, A, can not eqal zero. Therefore: sin k = 0 k = (A8) Sbstitting the vale of k into Eqation A8 and solving for the critical moment, ocr, ields: ocr EI GJ (A9) A.. INFINITE SERIES APPROXIATION If a beam has a constant cross-section bt the indced moment,, varies along the nbraced length,, a closed-form soltion does not exist and it mst be solved b other means. One techniqe is to se an infinite series approximation. For example, consider a cantilevered beam with a concentrated load at the end of the beam. The indced bending moment,, can be denoted as = P(-z) where z is measred from the spported end. B defining a new vale x = (-z) and a constant k = P /EI GJ, Eqation A4 can be rewritten as: d k dx x 0 (A0) To solve the second-order differential eqation in Eqation A0, assme that the shape of the crve for is a series of the form: a o 3 n ax ax a3x... an x... (A) Differentiating Eqation A twice, ields the following: d a dx 6a x a x 3 4 0a x n(n -)a n x n-... (A)

28 AERIAN FOREST & PAPER ASSOIATION TEHNIA REPORT NO. 4 3 Sbstitting Eqations A and A into Eqation A0 ields: [a 6a x...n(n -)a x ] [a k x a k x...a n- 3 n 3 n 0 n k x ] 0 (A3) ombining like terms ields the following eqation: 3 n- a o k a x a k 0a x... a k n(n -)a x 0 a 6a 3x 4 5 n-4 n (A4) It can be seen b inspection that Eqation A4 can onl be tre if each term in the eqation eqals zero. Therefore, a =0, a 3 =0, and for n4, a n- 4 k + n(n-)a n =0. Solving for a n : a n an4k n( n ) (A5) Sbstitting different vales of n into Eqation A5 ields the following set of eqations to solve for each coefficient: n=4 a 4 = -a 0 k /(3 4) n=5 a 5 = -a k /(4 5) n=6 a 6 = -a k /(5 6) = 0 n=7 a 7 = -a 3 k /(6 7) = 0 n=8 a 8 = -a 4 k /(7 8) = a 0 k 4 /( ) n=9 a 9 = -a 5 k /(8 9) = a k 4 /( ) n=0 a 0 = -a 6 k /(9 0) = 0 n= a = -a 7 k /(0 ) = 0 In general, for n=4, 8,, 6, 0, 4... a n = a 0 (-k ) n/4 / [ (n-) n] for n=5, 9, 3, 7,, 5... a n = a (-k ) (n-)/4 / [ (n-) n] for n=6, 0, 4, 8,, 6... a n = 0 for n=7,, 5, 9, 3, 7... a n = 0 Sbstitting these coefficients into Eqations A and A and simplifing ields the following eqations: a o k x k x k x k x k x k x... a... x (A6) d a dx o k x k x k x k x... a k x k x (A7) Given that there is no rotation of the beam at the fixed end and no change in rotation at the free end, the bondar conditions for this loading case can be expressed as follows: z = 0: x = : 0 d 0 dx z = : x = 0: 0

29 4 DESIGNING FOR ATERA-TORSIONA STABIITY IN WOOD EBERS Sbstitting the second condition into Eqation A7, it can be seen that constant a =0. Sbstitting the first condition into eqation A6, redces it to the following: a o 4 k k k (A8) To provide a nontrivial soltion a o can t eqal zero. Therefore, the minimm vale of k where = 0 occrs at k = Sbstitting the vale of k and solving for the critical vale of P ields: 4.03 Pcr EI GJ (A9) Finall, b setting Eqation A9 eqal to the maximm moment eqation for this loading condition ( max =P) and simplifing ields: cr 4.03 EI GJ (A0) A..3 STRAIN ENERGY APPROXIATION ETHOD Another techniqe sed to approximate the critical bckling load of beams is the strain energ approximation method. When a beam bckles, the strain energ de to lateral deflection and twist is increased. The load which is applied to the beam is lowered, thereb decreasing the potential energ. The decrease in potential energ can then be set eqal to the increase in strain energ. For example, take a simpl-spported single span beam with a concentrated load at midspan. The increase in strain energ de to lateral deflection and twist can be denoted as follows: U EI d dz 0 dz GJ The decrease in potential energ can be denoted as follows: 0 d dz dz (A) d V P zdz (A) 0 dz Eqating the increase in strain energ given in Eqation A to the decrease in potential energ given in Eqation A ields the following: P 0 d d z dz EI dz GJ dz 0 dz 0 d dz dz (A3) Using Eqation A, the following relationship can be defined for this loading condition: Sbstitting this vale into Eqation A3 and combining like terms ields: AERIAN WOOD OUNI

30 TEHNIA REPORT NO. 4 5 d dz P EI z (A4) Sbstitting this vale into Eqation A3 and combining like terms ields: P 4EI 0 z dz GJ 0 d dz dz (A5) To estimate the critical vale of P, a good approximation of the rotated shape along the length is needed. For a simpl-spported single span beam, the shape can be approximated sing a trigonometric series: a z 3z 5z cos a3 cos a5 cos... (A6) Sbstitting the first term of this approximation into Eqation A5: P 4EI 0 z a cos z dz GJ 0 a z sin dz (A7) Integrating both sides of the eqation reslts in: P 4EI a a GJ 4 (A8) Solving for the critical vale of P ields: P 7.6 cr EI GJ (A9) Finall, b setting Eqation A9 eqal to the maximm moment eqation for this loading condition ( max =P/4) and simplifing ields: cr 4.9 EI GJ (A30) Note: The actal constant in Eqation A9 is 6.93 rather than 7.6; abot a.5% error. This difference can be redced to less than 0.% b adding a second term to the approximation of in Eqation A7. AERIAN FOREST & PAPER ASSOIATION

31 6 DESIGNING FOR ATERA-TORSIONA STABIITY IN WOOD EBERS A..3 EQUIVAENT OENT FATOR The strain energ approximation method provides a tool for approximating the critical moment eqation for man loading conditions. Using the previos example and the maximm moment eqation for the previos loading condition ( max =P/4), Eqation A5 can be rewritten as: EI 0 z d dz GJ dz 0 dz (A3) Developing a similar eqation for the reference condition of a beam braced at both ends with a constant moment, Eqation A5 can be rewritten as: EI dz GJ o 0 0 d dz dz (A3) At this point an important assmption mst be clearl nderstood. The rotated shape,, is assmed to be approximatel the same for all beams braced at both ends. Therefore, the second term in both Eqations A3 and A3 are eqal. Setting these eqations eqal and sbstitting the single term approximation for (see Eqation A6) ields: EI 0 a dz EI z z o cos 0 z a cos dz (A33) Integrating both sides ields the following reslts: EI 0 ( 0.340a) (0.500a) EI (A34) Solving for the ratio, b, of the critical moment for the concentrated load case to the critical moment for the reference case ields: b cr ocr (A35) Note: The actal moment ratio for these two loading conditions is.347 rather than.366; abot a.5% error. This difference is de to the error associated with the assmed shape of. Eqation A33 can be solved sing an assmed shape of. The accrac of the assmed shape determines the error of the approximation. One commonl sed shape of is: a 4 z (A36) Sbstitting this vale for into Eqation A33 ields: AERIAN WOOD OUNI

32 TEHNIA REPORT NO. 4 7 a 4EI 0 4 z o a dz 4EI z 0 4 z dz (A37) Integrating both sides ields the following reslts: a 4EI 5 9 o a EI 5 60 (A38) Solving for the ratio, b, of the critical moment for the concentrated load case to the critical moment for the reference case ields: 3360 b.390 o 740 Note: For this load case, the moment ratio error is more than 3% as a reslt of the assmed shape of. (A39) It can be seen in the previos sections that the moment ratio, b, is a fnction of the area nder the prodct crve. The familiar for-moment empirical eqation was developed sing this relationship. Formlation of the for-moment eqation can be demonstrated sing the previos load case of a concentrated load at midspan and the Eqation A36 approximation of. The rotated shape, i, and the moment, i, at each qarterpoint location for a concentrated load at midspan are: 3a A 64 4a B 64 P A A (A40) 8 max P B B. 00 (A4) 4 max 3a 64 P max (A4) The vales of i for the niform moment load case are the same as those given in Eqations A40- A4; however, the moment is constant ( A = B = = o ). A constant,, is sed to calibrate the eqation which relates the prodcts for the two load cases as follows: b o A B A A B B max max max 3 A max 0 4 B 3 max max (A43) The constant,, sed to calibrate Eqation A43 varies with the loading condition. For the case of a concentrated load at midspan, the theoretical vale of b =.347 reslts in a constant =.65. For the load case of eqal and opposite end moments (doble crvatre), b =.30 reslting in a constant =.4. onservativel, the vale of for se with the empirical method for determining the eqivalent moment factor has been set at.5 reslting in the familiar eqation: AERIAN FOREST & PAPER ASSOIATION

33 8 DESIGNING FOR ATERA-TORSIONA STABIITY IN WOOD EBERS b 3 A B max.5 max (A44) This empirical procedre does not appl to beams braced at onl one end, sch as cantilevers. A. OAD EENTRIITY FATOR Eqations derived in A. assme that loading is throgh the netral axis of the member. In cases where the load is applied above the netral axis at an nbraced location, an additional moment is indced de to the added displacement of the load at the top of the beam. The effect of this additional moment on the critical bckling moment can be estimated sing the strain energ approximation method addressed in the previos section. Using the example from the previos section, a concentrated load is applied at midspan of a simpl-spported single span beam. The additional moment de to the load being applied at the top of the beam lowers the load point thereb decreasing the potential energ. The vertical distance, b, that the load is lowered can be denoted as follows: h h o b ( cos o ) (A45) 4 The decrease in potential energ can be denoted as: V Ph o 4 (A46) The location of the load is at midspan where z = 0. Using the previos definition of, Eqation A46 can be simplified to: Ph z Pha V a cos (A47) 4 4 Adding the decrease in potential energ from the load applied at the top of the beam to Eqation A7 ields: Pha P z a cos sin z a z dz GJ dz (A48) EI Integrating both sides of the eqation reslts in: Pha 4 a 94.6 a GJ 3 P 4EI 4 (A49) Eqation A49 can be solved for the vale P; however, it can be seen that Eqation A8 and Eqation A49 have similar terms and it can be seen that, after some mathematical maniplation, the can be eqated as follows: AERIAN WOOD OUNI

34 TEHNIA REPORT NO. 4 9 a GJ 4 Pna 4EI a Ptpha GJ 4 4 P tp 4EI (A50) Simplifing and solving for the critical load applied at the top of the beam, P tp, reslts in: P tp Pna h GJ P na Pna h GJ (A5) Dividing Eqation A5 throgh b P na ields: e P P tp na Pna h GJ Pna h GJ (A5) Terms in Eqation A5 can be rearranged to: e P P crtp crna (A53) where: P na h GJ (A54) Sbstitting the eqation for P na from Eqation A9 into Eqation A54 reslts in:.74h EI GJ (A55) Using the more accrate constant of 6.93 rather than 7.6 reslts in:.7h EI GJ (A56) Flint [0] demonstrated, nmericall, that Eqation A53 is accrate for cases where lies between and +.7 (negative vales indicate loads applied below the netral axis and positive vales indicate loads applied above the netral axis). imiting to a maximm vale of.7 wold limit e to a minimm vale of 0.7. For other loading conditions of a beam between spports, a similar procedre can be sed to determine. In general, Eqation A56 can be rewritten as follows: P na 4 V h Pha GJ (A57) AERIAN FOREST & PAPER ASSOIATION

35 30 DESIGNING FOR ATERA-TORSIONA STABIITY IN WOOD EBERS To demonstrate the application of Eqation A57, assme seven eqall spaced loads along a beam braced at both ends. Using Eqation A47, the effect of the mltiple loads can be denoted as: V Pha 4Pha cos 0.00 cos 0.5 cos 0.50 cos (A58) For loading throgh the netral axis, the eqation for calclating P cr for this loading condition is: P 3.55 cr EI GJ (A59) Sbstitting the vale for V and P cr into Eqation A57, the vale of for this loading condition can be estimated as:.44h EI (A60) GJ AERIAN WOOD OUNI

36 TEHNIA REPORT NO. 4 3 Appendix B Derivation of 997 NDS Beam Stabilit Provisions Otlined in this Appendix is the process b which the effective length provisions contained in the 00 and earlier versions of the NDS were developed b Hoole and adsen [7]. Using Eqation adjsted for the loading condition of a concentrated load on a simpl-spported beam: cr 4.3 e EI GJ (B) For rectanglar cross-sections, Hoole and adsen assmed G = E/6 : J = 4I (-0.63b/d) : = -(b/d) (see Appendix. for explanation) and expanded Eqation B to the following: cr.eei 0.63b / d ( b / d) (B) Recognizing that the term nder the radical reaches a minimm vale of when b/d = (see Appendix.), Eqation B was redced to: cr.0ei e (B3) Several approximation methods for determining the effect of load being applied at the top of the beam were reviewed. Hoole and adsen sed a std b Flint and, sing Timoshenko s first order energ approximation (corrected in a rederivation b Flint), derived simple effective length adjstments for the above mentioned load case. for (B4) e where:.733d EI GJ (B5) Note: The estimated vale of e in Eqation B4 is onl accrate in a limited range. Flint proposed a more accrate estimate of e sing a qadratic soltion (see Appendix A.) which was not sed b Hoole and adsen. The more accrate qadratic soltion is tilized in the procedres proposed in..3. and Hoole and adsen sed several approximations to simplif Timoshenko s first order approximation of e. The steps are covered in the following eqations: AERIAN FOREST & PAPER ASSOIATION

37 3 DESIGNING FOR ATERA-TORSIONA STABIITY IN WOOD EBERS e.733 d b d b d.733 d b 3d (B6) e d b.733 (B7) 3d d b.733 3d e d.733 b 3d 3d.9 (B8) Hoole and adsen combined Eqations B3 and B8 to prodce the final eqation: cr.4ei.9 3d (B9) The eqation was frther simplified to a general form for other loading conditions sing a term for effective length, e :.4EI cr (B0) For a concentrated load at midspan: e.9 e 3 d (B) For design, Hoole and adsen recommended that the nbraced length,, be increased 5% to accont for imperfect torsional restraint at the spports:.37 e 3 d (B) In 977, Eqations B0 and B were incorporated into the NDS. Additional effective length eqations were also developed for common loading conditions. In 986, the se of Eqation B was limited to.80 which occrs where /d 7:.8.37 e 3 d (B3) For se in the NDS, Eqation B0 was converted to a critical bckling design stress b dividing the eqation b the strong-axis section modls, S x, and mltipling b a series of adjstment factors to adjst the tablated average E vales to allowable 5 th percentile shear-free E vales. The final eqation took the form of: AERIAN WOOD OUNI

38 TEHNIA REPORT NO F be K be RB E (B4) where: K be OV E (B5) and, R B ed b (B6) AERIAN FOREST & PAPER ASSOIATION

39 34 DESIGNING FOR ATERA-TORSIONA STABIITY IN WOOD EBERS AERIAN WOOD OUNI

40 TEHNIA REPORT NO Appendix Derivation of Rectanglar Beam Eqations Otlined in this Appendix are assmptions made and procedres sed to develop the simplified beam stabilit eqations for rectanglar members.. RITIA BUKING OENT The general eqation for the critical bckling moment, cr, is as follows: E I G J b e cr ().5 For wood members, the E/G ratio ranges from to 3. In practice, a vale of 6 has tpicall been sed. For rectanglar cross-sections, the torsional shear constant, J, and the cross-section slenderness factor,, can be estimated as: 5 b b J 4I () d d b d Sbstitting these vales into Eqation, ields the following soltion: (3) cr E b e.3 05 I b b d d b d 5 (4) The term nder the radical in Eqation 4 reaches a minimm vale of when b/d = This can be seen in the following graph: Ratio b/d AERIAN FOREST & PAPER ASSOIATION

41 36 DESIGNING FOR ATERA-TORSIONA STABIITY IN WOOD EBERS Sbstitting this minimm vale into Eqation 4 ields: cr.3 bee 05 I (5). OAD EENTRIITY FATOR FOR TOP-OADED BEAS The general eqation for the load eccentricit factor for top-loaded beams, e, is as follows: where: e (6) kd E 05I (7) G J 05 Using the relationships assmed in the previos section for rectanglar wood members, the term can be simplified as follows: kd b b d d 5 b d (8) The term nder the radical in Eqation 8 increases as the b/d ratio of the member increases and, therefore, redces the critical load. This can be see in the following graph: Ratio b/d A maximm b/d ratio was chosen as representing the dimensions of a x4. For larger b/d ratios, this estimate is nconservative. However, the critical bckling moment for these cases is increasing at a rate that makes this adjstment insignificant. At b/d=0.486, Eqation 8 redces to:.3kd (9) AERIAN WOOD OUNI

42 TEHNIA REPORT NO Appendix D Graphical Representation of Wood Beam Stabilit Provisions (for beams loaded throgh the netral axis) oading ondition Eqivalent oment Factor, b Uniforml distribted load oncentrated center Two eqal conc. /3 points Three eqal conc. /4 points For eqal conc. /5 points Five eqal conc. /6 points Six eqal conc. /7 points Seven eqal conc. /8 points Eight eqal conc. /9 points Spported at oading Point Single Span Beams SSSSSSSSSSSSSSSSSSSSSSSSSSSS SSSSSSSSSSSSSSSSSSSSSSSSSSSS SSSSSSSSSSSSSSSSSSSSSSSSSSSS SSSSSSSSSSSSSSSSSSSSSSSSSSSS SSSSSSSSSSSSSSSSSSSSSSSSSSSS SSSSSSSSSSSSSSSSSSSSSSSSSSSS SSSSSSSSSSSSSSSSSSSSSSSSSSSS SSSSSSSSSSSSSSSSSSSSSSSSSSSS Unspported at oading Point SSSSSSSSSSSSSSSSSSSSSSSSSSSS.3 SSSSSSSSSSSSSSSSSSSSSSSSSSSS.35 SSSSSSSSSSSSSSSSSSSSSSSSSSSS.4 SSSSSSSSSSSSSSSSSSSSSSSSSSSS.4 SSSSSSSSSSSSSSSSSSSSSSSSSSSS.4 SSSSSSSSSSSSSSSSSSSSSSSSSSSS.4 SSSSSSSSSSSSSSSSSSSSSSSSSSSS.3 SSSSSSSSSSSSSSSSSSSSSSSSSSSS.3 SSSSSSSSSSSSSSSSSSSSSSSSSSS.3 Eqal end moments (opposite rotation) SSSSSSSSSSSSSSSSSSSS.00 Eqal end moments (same rotation) oncentrated nspported end Uniforml distribted load antilever Beams SSSSSSSSSSSSSSSSSSSS.30 SSSSSSSSSSSSSS.8 SSSSSSSSSSSSSSSSS.05 AERIAN FOREST & PAPER ASSOIATION

43 American Forest & Paper Association American Wood oncil 9th Street, N.W., Site 800 Washington, D awcinfo@afandpa.org America s Forest & Paper People Improving Tomorrow s Environment Toda -03